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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Classical mechanics as an example of generalized quantum mechanics

K. R. W. Jones

Published Apr 15, 1992• 13 citations
Publisher

Abstract

It is shown how a simple energy functional ansatz within Weinberg's theory of generalized quantum mechanics reduces the full functional dynamics to an effective classical dynamics governed by the equations of Hamilton.
DOI
10.1103/physrevd.45.r2590

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The possibility of generalizing quantum mechanics in such a way as to retain its predictive results, while comprehending additional solutions, is examined. It is found that this can be done through a perfected formal correspondence with Hamilton-Jacobi mechanics, by which one is led to consider generalizations of the Heisenberg postulate of the form ${p}_{k}{q}_{j}\ensuremath{-}{q}_{j}{p}_{k}=S{\ensuremath{\delta}}_{\mathrm{jk}}$, where SSS is a quantum analog of Hamilton's principal function. The formalism is shown to be equivalent to a simple change in Hamiltonian, with transformed momentum operators satisfying conventional commutation relations, and with an additional relationship involving formal analogs of the classical "initial constants" adjoined. A particular choice of $S(=\frac{\ensuremath{\hbar}}{i})$ leads to a theory identical with wave mechanics a part from a constant (unobservable) phase factor on the wave function. The fact that SSS may possess other, nonconstant values, demonstrated by a specific example, suggests the ability of the mechanical equations to describe a broader class of physical states than has hitherto been investigated.
The classical limit ℏ\hbarℏ->0 of quantum mechanics is known to be delicate, in particular there seems to be no simple derivation of the classical Hamilton equation, starting from the Schrödinger equation. In this paper I elaborate on an idea of M. Reuter to represent wave functions by parallel sections of a flat vector bundle over phase space, using the connection of Fedosov's construction of deformation quantization. This generalizes the ordinary Schrödinger representation, and allows naturally for a description of quantum states in terms of a curve plus a wave function. Hamilton's equation arises in this context as a condition on the curve, ensuring the dynamics to split into a classical and a quantum part.
Negative energy states are applied in Krein space quantization approach to achieve a naturally renormalized theory. For example, this theory by taking the full set of Dirac solutions, could be able to remove the propagator Green function's divergences and automatically without any normal ordering, to vanish the expected value for vacuum state energy. However, since it is a purely mathematical theory, the results are under debate and some efforts are devoted to include more physics in the concept. Whereas Krein quantization is a pure mathematical approach, complex quantum Hamiltonian dynamics is based on strong foundations of Hamilton-Jacobi (H-J) equations and therefore on classical dynamics. Based on complex quantum Hamilton-Jacobi theory, complex spacetime is a natural consequence of including quantum effects in the relativistic mechanics, and is a bridge connecting the causality in special relativity and the non-locality in quantum mechanics, i.e. extending special relativity to the complex domain leads to relativistic quantum mechanics. So that, considering both relativistic and quantum effects, the Klein-Gordon equation could be derived as a special form of the Hamilton-Jacobi equation. Characterizing the complex time involved in an entangled energy state and writing the general form of energy considering quantum potential, two sets of positive and negative energies will be realized. The new states enable us to study the spacetime in a relativistic entangled "space-time" state leading to 12 extra wave functions than the four solutions of Dirac equation for a free particle. Arguing the entanglement of particle and antiparticle leads to a contradiction with experiments. So, in order to correct the results, along with a previous investigation [1], we realize particles and antiparticles as physical entities with positive energy instead of considering antiparticles with negative energy. As an application of modified descriptions for entangled (space-time) states, the original version of EPR paradox can be discussed and the correct answer can be verified based on the strong rooted complex quantum Hamilton-Jacobi theory [2-27] and as another example we can use the negative energy states, to remove the Klein's paradox without the need of any further explanations or justifications like backwardly moving electrons. Finally, comparing the two approaches, we can point out to the existence of a connection between quantum Hamiltonian dynamics, standard quantum field theory, and Krein space quantization [28-43].
This work explores the quantum propagator K(x,t)K(x,t)K(x,t) as a solution of the system's dynamical equation. We develop a generalized propagator framework in which the propagator is written in the form K=exp⁡ ⁣(R+iℏS)K=\exp\!\left(R+\frac{i}{\hbar}S\right)K=exp(R+ℏi​S), where SSS governs the semiclassical phase structure and RRR governs the amplitude transport and weighting of configurations. Starting from nonrelativistic quantum mechanics, the classical Hamilton--Jacobi equation emerges in the semiclassical limit, while RRR reduces to the logarithm of the Van Vleck determinant. The formalism is then extended to relativistic field theory and minisuperspace quantum cosmology using functional methods and generalized Hamiltonian constraints. Then the resulting semiclassical equations recover the corresponding classical dynamics along characteristic flows. In minisuperspace models, the formalism yields coupled equations for geometric and matter sectors analogous to semiclassical Wheeler--DeWitt systems. In semiclassical gravitational settings, the gravitational contribution may admit an entropy-like interpretation, consistent with thermodynamic weighting factors appearing in Euclidean gravitational path integrals. The resulting framework suggests a unified semiclassical description in which propagator phases encode dynamical evolution while amplitude functionals encode transport and configuration weighting. This perspective motivates viewing mechanics as a relation between the differential structure of spacetime and the differential structure of the space of dynamical configurations.